On the quadratic Fock functor

TL;DR

This paper characterizes when the quadratic second quantization of an operator on certain function spaces results in an orthogonal projection, showing it occurs only for multiplication by characteristic functions.

Contribution

It provides a necessary and sufficient condition for quadratic second quantization to be an orthogonal projection, specifically identifying multiplication by characteristic functions as the unique case.

Findings

Quadratic second quantization of an operator is an orthogonal projection only if the operator is multiplication by a characteristic function.

The result characterizes the structure of operators that preserve projection properties under quadratic second quantization.

The paper establishes a precise criterion linking operator form to the projection property in quadratic Fock space.

Abstract

We prove that the quadratic second quantization of an operator p on $L^2(\mathbb{R}^d)\cap L^\infty (\mathbb{R}^d)$ is an orthogonal projection on the quadratic Fock space if and only if p =MI, where MI is a multiplication operator by a characteristic function I.